Here’s a list of 100 chapter titles for a book or course on Homological Algebra, progressing from beginner to advanced topics with a focus on mathematical concepts:
Advanced Techniques in the Study of Derived Categories
The Use of Derived Functors in Algebraic Topology
The Structure of Derived Categories and Their Applications
Applications of Homological Algebra in Category Theory
Advanced Spectral Sequences and Their Computations
The Grothendieck Spectral Sequence
Homotopy Categories and Derived Categories
Homological Methods in Stable Homotopy Theory
The Bousfield-Kan Spectral Sequence
Introduction to Triangulated Categories
The Cotangent Complex and its Homological Properties
The Theory of Infinity-Categories and its Relation to Homological Algebra
Derived Functors in the Context of Higher Categories
Derived Categories of Schemes and Their Applications
Homotopical Algebra: Background and Results
The Dold-Thom Theorem and Its Connection to Homology
The Poincaré Duality in Homology and Cohomology
Advanced Applications of Homology in Algebraic Geometry
The Theory of Group Cohomology in Homological Algebra
Homological Algebra in the Study of Lie Algebras
The Use of Homological Algebra in Model Categories
Derived Categories and Localization in Algebra
Morita Equivalence and Homological Algebra
Advanced Topics in Local Cohomology
Homology with Infinite-Dimensional Categories
Applications of Derived Categories in String Theory
Compact Objects and Their Homological Properties
The Role of Homological Algebra in Homotopy Theory
The Classification of Derived Categories and Their Applications
Open Problems and Future Directions in Homological Algebra
These chapters take the reader through Homological Algebra from foundational concepts such as exact sequences, projective and injective modules, and tensor products, to advanced topics like derived categories, spectral sequences, and applications in algebraic geometry and topology.